Valuation of investment activity taking into account the ratio of profitability and risk
Obviously, each investor wants his manager's efforts to lead to the formation of a portfolio that would yield returns higher than unregulated portfolios, such as a market portfolio (estimated on the basis of some market index, say, the MICEX index). Strictly speaking, such a result can be achieved either by selection (passive management method) - selection in a diversified portfolio of securities having a better than average market risk/yield ratio, or by timing (active management) - a constant sale of financial assets whose price decreases, and purchases of financial assets, growing in price, or both ways simultaneously. Note that the evaluation of the results of portfolio management only on the basis of the realized profitability of the regulated portfolio is not entirely correct, since it does not take into account the risk of investing in the securities portfolio. So, if the portfolio of the fund includes mainly securities with fixed income, then the yield of such a portfolio is certainly lower than the yield of the fund's portfolio operating on shares; consequently, the results of portfolio management of funds can not be assessed only on the achieved portfolio returns.All this forces us to find ways of evaluating investment activities that take into account the ratio of yield and risk of the securities portfolio. The methods of determining the optimal portfolio, considered earlier, make it possible to develop criteria for the evaluation of investment activity on the basis of the CAPM theory and to introduce special units, measures, such estimates that take into account both the realized profitability and the risk of investment.
When using the CAPM model to estimate the ratio of yields and portfolio risk for past periods of calculation, it is necessary to take into account that the CAPM theory allows us to explain the pricing of financial assets using the expected future yields and risks. In other words, the CAPM model describes how the price of financial assets should be set:
However, if the expected yields for CAPM are not too different from the actual observed data and if the return on financial assets for the past and future periods do not have significant variances, then the measures based on the CAPM can be used to assess investment activity for period. The most known measures of this evaluation are the measures proposed by J. Treynor ( Jack Treynor), M. Jensen ( Michael Jensen) and W. Sharpe ( William Sharpe).
Measure Traynor
Traynor proposed to introduce a special unit, designated by him 7), as a measure that estimates, on the basis of the data already observed, the share of realized portfolio returns per unit of systematic risk. At the same time, the degree of systematic risk is estimated by the coefficient beta portfolio:
where - the average value of the observed portfolio returns for the period under study (the average yield can be chosen as the arithmetic mean or geometric mean); - the average value of observed risk-free returns for the period under study; - Portfolio beta of the assessed portfolio.
The measure of any i portfolio should be compared with the market measure equal to , since the beta quotient market portfolio is equal to one. If for the portfolio in question , then this portfolio outperforms the market and portfolio management results should be recognized as positive. If, however, , then the portfolio was managed unsatisfactorily.
The algorithm for evaluating the results of portfolio management using the Traynor measure is as follows.
1. Select the time interval in the past that is being evaluated, and divide this interval by N equal in duration of calculation steps.
2. Given the available data on the value of the portfolio at the beginning and at the end of each calculation step, as well as the observed cash flows, calculate, for one of the methods of portfolio return r i, t, considered in the previous paragraph, for the calculation steps.
3. Select the index of the securities market and on its basis calculate the yield of the market portfolio for the same calculation steps.
4. Using the values and calculate the value of the portfolio bet for the estimated portfolio.
5. To allocate the risk-free return values for each calculation step.
6. Find the mean values
7. Calculate the Traynor measures for the estimated portfolio and for the market portfolio, then compare them and draw the appropriate conclusion about the quality of portfolio management.
Imagine that the activity of the three managers managing the portfolios A, B , C for the last three years is estimated. Based on the available information on the annual returns of portfolios, we calculate the average geometric annual return of each portfolio . Knowing the changes in the market index for the same period of time, we calculate the portfolio portfolio of each portfolio, and also define the average geometric annual values of the risk-free interest rate and the yield of the market portfolio (conditional data). Suppose that the following data are obtained (Table 5.4).
Table 5.4
Data for the calculation of the Traynor measure
Options |
Portfolios |
Market |
||
A |
In |
C |
||
Average return |
0.124 |
0.136 |
0.118 |
0.108 |
Beta quotient |
1,120 |
0.820 |
1.450 |
1,000 |
Traynor measure |
T A = 0.060714 |
T B = 0.097561 |
T C = 0.042759 |
T M = 0.052 |
The values of T i for the estimated portfolios are calculated as follows:
As follows from Table. 5.4, from the point of view of realized profitability, all three portfolios outperform the market portfolio. However, these returns must also be correlated with the level of systematic risk (beta coefficient) of each portfolio. For a portfolio С the risk is incommensurably high in comparison with the average realized profitability , therefore the market portfolio dominates the portfolio С.
Using the Traynor measure can also be illustrated graphically (Figure 5.1).
Fig. 5.1. Graphic interpretation of the Traynor measure
The graphs of the points A, B, C and M correspond to the mean values: and ; and ; and ; and img src="images/image832.jpg"> Through these points and the point the lines that Traynor named lines of possible portfolios ( portfolio opportunity line - PPL ). They show what composite portfolios can be formed by combining risk-free lending or loans and portfolios A, B , C.
Through the M line the SML line is drawn. Let's pay attention to the fundamental difference between SML, constructed for the expected values , and SML, based on the results of actual, realized data. In the first case, all securities and all portfolios correspond to SML points, otherwise they will be incorrectly estimated and market mechanisms will lead prices to equilibrium. When SML is built on the basis of actual data, portfolios and securities may not lie on the SML, line, as in our case for A, B and c.
PPL portfolio B dominates all other lines, so you can argue that the manager of this fund has achieved the best results over the past three years. The worst result is portfolio manager C, since for this portfolio the resulting risk/yield ratio is below market.
A specific case occurs when the portfolio "beta" managed portfolio is negative (). Obviously, in this case, when the Traynor measure is negative , therefore, it is always less . However, on this basis, it can not be said that this managed portfolio is worse than the market one. To evaluate the managed i yurtfel in the case of , you should proceed as follows:
a) find the value of the average portfolio return, corresponding to the CAPM model, with
b) compare the previously calculated value of and . If , then the managed portfolio is better than the market one, and the portfolio management results can be considered satisfactory.
thematic pictures
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